Physlib.Particles.StandardModel.Fermions.UpSinglet

The definition establishes a linear equivalence between the type `UpSinglet`, which represents the target vector space of an up-type singlet quark field in the Standard Model, and its underlying representation as the tensor product $\text{RightHandedWeyl} \otimes_{\mathbb{C}} \mathbb{C}^3$. Here, $\text{RightHandedWeyl}$ is the space of right-handed Weyl fermions and $\mathbb{C}^3$ (represented as `EuclideanSpace ℂ (Fin 3)`) is the 3-dimensional complex color space.

The space `UpSinglet`, which represents the target vector space of an up-type singlet quark field in the Standard Model, is equipped with the structure of an additive commutative group. This structure is inherited from the underlying representation $\text{RightHandedWeyl} \otimes_{\mathbb{C}} \mathbb{C}^3$ via the equivalence `valEquiv`.

The space `UpSinglet`, which represents the target vector space of an up-type singlet quark field in the Standard Model, is equipped with the structure of a module over the complex numbers $\mathbb{C}$. This complex vector space structure is inherited from its underlying representation as the tensor product $\text{RightHandedWeyl} \otimes_{\mathbb{C}} \mathbb{C}^3$ (where $\text{RightHandedWeyl}$ is the space of right-handed Weyl fermions and $\mathbb{C}^3$ is the color space) via the equivalence `valEquiv`.

The linear equivalence $valLinEquiv$ provides an isomorphism between the space of up-type singlet quarks $\text{UpSinglet}$ and its underlying representation as the tensor product $\text{RightHandedWeyl} \otimes_{\mathbb{C}} \mathbb{C}^3$. Here, $\text{RightHandedWeyl}$ denotes the space of right-handed Weyl spinors and $\mathbb{C}^3$ (represented by $\text{EuclideanSpace } \mathbb{C} \text{ (Fin 3)}$) denotes the color space of the Standard Model.

Let $q$ be an element of the space $\text{UpSinglet}$, which represents the target vector space of an up-type singlet quark field in the Standard Model. Let $\text{valLinEquiv} : \text{UpSinglet} \simeq_{\mathbb{C}} \text{RightHandedWeyl} \otimes_{\mathbb{C}} \mathbb{C}^3$ be the linear isomorphism between the up-type singlet space and its underlying representation as the tensor product of right-handed Weyl fermions and the color space $\mathbb{C}^3$. The theorem states that applying the map $\text{valLinEquiv}$ to $q$ is equal to accessing its underlying value $q.\text{val}$ in the tensor product space: $$\text{valLinEquiv}(q) = q.\text{val}$$

Let $m \in \text{RightHandedWeyl} \otimes_{\mathbb{C}} \mathbb{C}^3$ be an element in the tensor product space of right-handed Weyl fermions and the Standard Model color space. The inverse of the linear equivalence $\text{valLinEquiv} : \text{UpSinglet} \simeq_{\mathbb{C}} \text{RightHandedWeyl} \otimes_{\mathbb{C}} \mathbb{C}^3$, when applied to $m$, is equal to the element $\langle m \rangle$ in the $\text{UpSinglet}$ space.

Let $q_1$ and $q_2$ be elements of the space `UpSinglet`, which represents the target vector space of up-type singlet quark fields in the Standard Model. Let $\text{val}$ be the map that extracts the representation of these fields in the underlying tensor product space $\text{RightHandedWeyl} \otimes_{\mathbb{C}} \mathbb{C}^3$. Then, the addition of two fields in `UpSinglet` is compatible with the addition in the underlying space, such that $(q_1 + q_2).\text{val} = q_1.\text{val} + q_2.\text{val}$.

Let $r \in \mathbb{C}$ be a complex scalar and $q \in \text{UpSinglet}$ be an up-type singlet quark field. Let $\text{val}$ be the map that extracts the representation of the field in the underlying tensor product space $\text{RightHandedWeyl} \otimes_{\mathbb{C}} \mathbb{C}^3$. Then, the scalar multiplication of a field in $\text{UpSinglet}$ is compatible with the scalar multiplication in the underlying space, such that $\text{val}(r \cdot q) = r \cdot \text{val}(q)$.

The complex representation of the group $SL(2, \mathbb{C})$ (the double cover of the restricted Lorentz group) on the space of up-type singlet quark fields $\text{UpSinglet}$. Under the linear isomorphism $\text{UpSinglet} \cong \text{RightHandedWeyl} \otimes_{\mathbb{C}} \mathbb{C}^3$, the action of an element $\Lambda \in SL(2, \mathbb{C})$ is defined as the tensor product of the right-handed Weyl representation on the spinor factor and the trivial representation on the 3-dimensional color space $\mathbb{C}^3$.

The representation of the Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ on the space of up-type singlet quark fields $\text{UpSinglet}$. Identifying $\text{UpSinglet}$ with the tensor product $\text{RightHandedWeyl} \otimes_{\mathbb{C}} \mathbb{C}^3$, the action of an element $g \in \mathcal{G}$ on a pure tensor $\psi \otimes v$ is given by: $$g \cdot (\psi \otimes v) = (g_{U(1)})^4 \psi \otimes (g_{SU(3)} v)$$ where $g_{U(1)} \in U(1)$ is the projection of $g$ onto the $U(1)$ factor and $g_{SU(3)} \in SU(3)$ is the projection of $g$ onto the $SU(3)$ factor. The $SU(2)$ component of the gauge group acts trivially.

Let $g$ be an element of the Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$. For a right-handed Weyl spinor $\psi$ and a color vector $v \in \mathbb{C}^3$, the representation of $g$ acting on the pure tensor $\psi \otimes v$ in the space of up-type singlet quark fields $\text{UpSinglet}$ is given by: $$g \cdot (\psi \otimes v) = (g_{U(1)})^4 \psi \otimes (g_{SU(3)} v)$$ where $g_{U(1)} \in U(1)$ and $g_{SU(3)} \in SU(3)$ are the components of $g$ in their respective subgroups.

Let $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group. Let $g \in \mathcal{G}$ be an element, and let $g_{U(1)} \in U(1) \subset \mathbb{C}$ and $g_{SU(3)} \in SU(3)$ be its projections onto the $U(1)$ and $SU(3)$ factors, respectively. Let $\{e_k\}_{k \in \{0, 1\}}$ be the basis for the space of right-handed Weyl fermions and $\{c_i\}_{i \in \{0, 1, 2\}}$ be the standard basis for the color space $\mathbb{C}^3$. The action of the representation of the gauge group on the basis element $e_k \otimes c_i$ of the up-type singlet quark space $\text{UpSinglet}$ is given by the sum: $$g \cdot (e_k \otimes c_i) = \sum_{i' = 0}^2 (g_{U(1)}^4 \cdot (g_{SU(3)})_{i'i}) (e_k \otimes c_{i'})$$ where $(g_{SU(3)})_{i'i}$ is the entry of the $SU(3)$ matrix at row $i'$ and column $i$.

Let $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group. For any two elements $g_1, g_2 \in \mathcal{G}$, the linear operators $\rho(g_1)$ and $\rho(g_2)$ representing their action on the space of up-type singlet quark fields are equal if and only if for all matrix indices $i, j$, the following equality holds: $$(g_1)_{U(1)}^4 ((g_1)_{SU(3)})_{ij} = (g_2)_{U(1)}^4 ((g_2)_{SU(3)})_{ij}$$ where $(g)_{U(1)}$ is the projection of $g$ onto the $U(1)$ component (represented as a complex number of unit norm) and $((g)_{SU(3)})_{ij}$ is the $(i, j)$-th entry of the $3 \times 3$ matrix representing the $SU(3)$ component of $g$.

Let $g$ be an element of the Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$, and let $\rho$ be the representation of $\mathcal{G}$ on the space of up-type singlet quark fields $\text{UpSinglet}$. Then $g$ lies in the kernel of $\rho$ if and only if there exists a complex number $a \in \mathbb{C}$ such that the $SU(3)$ component of $g$, denoted $g_{SU(3)}$, is the scalar matrix $a I$ (where $I$ is the identity matrix) and the $U(1)$ component of $g$, denoted $g_{U(1)}$, satisfies $a \cdot g_{U(1)}^4 = 1$.

Let $G_I = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group without discrete quotients. Let $\text{UpSinglet}$ be the vector space representing the up-type singlet quark field, and let $\rho$ be the representation of $G_I$ on $\text{UpSinglet}$. Then the central subgroup of $G_I$ associated with the $\mathbb{Z}_6$ quotient, denoted $\text{subgroup}(\mathbb{Z}_6)$, is contained in the kernel of the representation $\rho$ ($\ker \rho$).

For any choice of discrete quotient $Q$ of the Standard Model gauge group $G_I = SU(3) \times SU(2) \times U(1)$, the associated central subgroup $\text{subgroup}(Q) \subseteq G_I$ is contained within the kernel of the representation of $G_I$ acting on the space of up-type singlet quarks ($\text{UpSinglet}$).

Given a quotient parameter $Q$ of type `GaugeGroupQuot`, this definition defines a representation of the corresponding global Standard Model gauge group $\text{GaugeGroup}(Q)$ on the space of up-type singlet quarks $\text{UpSinglet}$ over the complex numbers $\mathbb{C}$. For the un-quotiented case ($Q = I$), the representation is given by `repGaugeGroupI`. For the cases where $Q$ corresponds to a discrete quotient (by $\mathbb{Z}_2$, $\mathbb{Z}_3$, or $\mathbb{Z}_6$), the representation is defined by lifting the representation of the un-quotiented group $G_I = SU(3) \times SU(2) \times U(1)$ to the quotient group $G_I / \Gamma_Q$. This lift is well-defined because the discrete subgroup $\Gamma_Q$ is contained in the kernel of the representation of $G_I$ on $\text{UpSinglet}$.